Crucial ECZ Exam Requirement
Circle theorems carry 6 to 8 marks in ECZ Grade 12 Mathematics. Never write an angle answer without stating the official geometric reason in brackets! You lose half the marks if reasons like "angle in a semicircle" or "angles in the same segment" are missing.
Table of Contents
1. The 7 Essential Circle Theorems
Theorem 1: Angle at the Centre
The angle subtended by an arc at the centre of a circle is twice the angle subtended by it at any point on the circumference: $\angle AOC = 2 \times \angle ABC$.
ECZ Reason: "Angle at centre is twice angle at circumference."
Theorem 2: Angle in a Semicircle
The angle in a semicircle is a right angle ($90^\circ$). If $AB$ is a diameter, any angle subtended on the circumference is $90^\circ$: $\angle APB = 90^\circ$.
ECZ Reason: "Angle in a semicircle."
Theorem 3: Angles in the Same Segment
Angles subtended by the same arc in the same segment of a circle are equal: $\angle ADB = \angle ACB$ (the "Bow-tie" theorem).
ECZ Reason: "Angles in the same segment."
Theorem 4: Cyclic Quadrilateral
A cyclic quadrilateral has all four vertices on the circumference. Opposite angles sum to $180^\circ$: $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
Exterior angle equals interior opposite angle.
ECZ Reason: "Opposite angles of a cyclic quadrilateral sum to 180°."
2. Tangent Theorems & Properties
Theorem 5: Tangent-Radius Perpendicularity
A tangent to a circle is perpendicular to the radius at the point of contact: $\angle OTP = 90^\circ$.
ECZ Reason: "Tangent is perpendicular to radius."
Theorem 6: Tangents from an External Point
The lengths of two tangents drawn from an external point $P$ to the points of contact $A$ and $B$ are equal: $PA = PB$, forming an isosceles triangle $\triangle PAB$.
ECZ Reason: "Tangents from an external point are equal."
3. Alternate Segment Theorem
Theorem 7: Alternate Segment Theorem
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment: $\angle TAB = \angle ACB$.
ECZ Reason: "Alternate segment theorem."
4. Worked Exam Past Paper Example
ECZ Past Examination Question:
In the circle with centre $O$, points $A, B, C, D$ lie on the circumference. Line $TA$ is a tangent to the circle at $A$. Given that $\angle TAB = 38^\circ$, $AC$ is a diameter, and $\angle ABD = 25^\circ$, calculate:
- $\angle ACB$
- $\angle ABC$
- $\angle BAC$
- $\angle ACD$
Model Answers with ECZ Marking Scheme Reasons:
- $\angle ACB = 38^\circ$ [Reason: Alternate segment theorem].
- $\angle ABC = 90^\circ$ [Reason: Angle in a semicircle, since AC is a diameter].
- $\angle BAC = 180^\circ - (90^\circ + 38^\circ) = 52^\circ$ [Reason: Angle sum of a triangle $\triangle ABC$].
- $\angle ACD = \angle ABD = 25^\circ$ [Reason: Angles in the same segment subtended by chord AD].
Geometry Compass Drill
Do not study circle theorems by reading alone. Use a mathematical drawing compass to construct circles, draw chords, tangents, and measure angles with a protractor to physically confirm each theorem.